## Archive for **August 9th, 2011**

## Doing Maths Without Numbers

Define ‘one’. Go on, try.

Well, ‘one’ is what you have when there’s only one of something. You have one duck. There – one. But that’s circular. You still need to define ‘one’ for you to have one duck.

All right, it’s one more than zero. ‘One’ more? Circular again, and what’s this ‘zero’ you’re talking about now?

Half of two? Okay, fine, but what’s two? Well, that’s twice one, isn’t it? Circular again. People trust mathematics with their lives, and yet all it seems to rest on is circular arguments. Can’t we do better than this?

Let’s start at the very beginning, with something really fundamental: “things are what they are.” A duck is a duck. A submarine is a submarine. “It is what it is” is a property shared by all things. Let’s call this the property of *identity*, of *being identical with itself –* the property of being what it is.

Everything is identical to itself. If you grouped together everything that is identical to itself, the set would have *everything *in it. And the set of everything *not* identical to itself would be empty.

We have defined an *empty set* – a group with no members. The set of four-sided triangles is empty, because nothing is both a triangle and four-sided. So is the set of things that are not identical to themselves, because nothing is.

Now we’re used to the idea of sets, consider this: what does the set of three ducks have in common with the set of three triangles? Call it the *cardinality* of the set – a measurement of how much is in it.

Imagine the set of all empty sets. What do all these empty sets have in common? Let’s call it *zero* – the cardinality of an empty set. We have now defined zero. If something is the same as zero, then it has the same cardinality as the set of all things not identical to themselves.

Now that we know what zero is, how many things are identical to zero? There’s just one thing that’s identical to zero: zero. If you filled a set with things identical to zero, there would only be one thing in it. The cardinality of the set of all things that are identical to zero, is one.

And the cardinality of the set of all things that are identical to ‘one’ or ‘zero’ is two. And the cardinality of the set of all things that are identical to ‘zero’ or ‘one’ or ‘two’ is three, and so on and so on and so on.

To break the circularity of mathematics, Bertrand Russell defined ‘I have two hands’ as:

“There is such an *a* such that there is a *b* such that *a* and *b* are not identical and whatever *x* may be, ‘*x* is a hand of mine’ is true when, and only when, *x* is *a* or *x* is *b*.”

And this is how you do maths without numbers.

REFERENCES

The BR quote was from *A History of Western Philosophy* (1945); also see *My Philosophical Development* (1959).

The layout of the above argument was from a YouTube video on mathematical logic, but I can’t remember where it is; I’ll update this if I find it.